Friday is food bank day at the Matthew Tree Project (MTP). A steady stream of clients files into the Bristol City mission. Mums with buggies, tall African men; they wait patiently on red plush pews that have been rearranged for the morning's activities. They check in with a counsellor, then go to the food store for groceries.

The clients are here because they are penniless; they exchange the voucher they have been issued with by a referring agency (referrers include local primary schools, Citizen's Advice, children's centres, asylum-seeker charities) for a bag of food collected by volunteers. So far, so like a standard food bank. But MTP's founder Mark Goodway is uncomfortable with the comparison. "We hate to be called a food bank," he says.

Goodway calls MTP a food store, which recognises that food offers a vital connection to people in financial hardship, but holds out a more demanding proposition: that simply handing out a food parcel does nothing to change the underlying poverty facing clients, particularly when impoverishment is not just a blip but something longer-term and more intractable.

Goodway is dismissive of what he calls the "food-bank mentality": the stigma food banks heap on clients; the way they can be more geared to satisfiying the spiritual needs of the provider rather than the material needs of the beneficiary; their inadequacy as a strategic response to poverty.

"If they [the client] go on Monday and get a three-day parcel, what happens on Thursday?" asks Goodway. "You don't solve the problems, it makes no sense."

Programme of support

Unlike many food banks, MTP puts no block on the number of food parcels clients can receive (other food banks limit clients, in theory, to three vouchers in one year. Clients sign up to a programme of support, with goals and an agreed exit point (when they have a job, say, or enough financial stability).

They receive counselling, debt advice and health advice each time they visit. Some clients receive an outreach service; others are signed up to a food skills and cookery course. Every six weeks they have to be re-referred by the agency that sent them. The emphasis is on taking personal responsibility, building life skills and breaking dependency on benefits and charity, to enable them to live "dignified, self-sufficient lives".

Clients may arrive in an emergency, but the ensuing intervention is designed to stabilise them and prevent further crisis. Once on the scheme, in theory, they stay as long as it takes, whether six weeks or six months. "Our job is to help them [the clients] believe in themselves, that they can have some degree of influence over what happens to them, gives them a sense of belief and hope," says Goodway.

The MTP's work was cited as "possible model for the capital" in a recent London assembly investigation into food poverty. Labour member Fiona Twycross, who led the investigation, said she was struck by its long-term approach to finding a route out of poverty for clients. "Food banks are not things we should expect in our society. But I liked the Matthew Tree ethos. It was substantially less disempowering than others we saw."

In Doncaster, they are also trying to do things differently. At Balby Bridge food bank there are nearly 30 blue carrier bags packed with food lined up at the back of St James' church hall. An hour and a half later they are all gone. This is a deprived area of south Yorkshire, and, while money has always been tight, welfare changes have pushed more people into poverty.

Church volunteers run the food bank, which is supplied by Food Aware, a local social enterprise that specialises in tackling food waste and food poverty. Anyone who takes food must pay £2.50, which is a fraction of the value of the groceries in the bag, but sends a message that this is not, strictly speaking, a charity food "handout".

Food Aware managing director Sean Gibbons is uneasy about what he calls "the food bank malarky". He has been developing food and nutrition skills projects for five years, long before food banks had any currency in the UK. The food bank name is a badge of convenience, he says, because it has resonance with the public and funders.

But it doesn't describe what he is trying to do, he says, which is to develop a food club or food co-op approach that recognises many clients' long-term issues can't be tackled by three food vouchers alone. "Take benefit sanctions," says Gibbons, They [the jobcentre] can now give you a six-month sanction. How many food parcels would that involve?"

He cites the case of a client he has been working with: "I'm trying to help him deal with things that are way beyond three food parcels. If he went to an ordinary food bank, would they help him tackle his Work Programme manager, or write to his MP? Forget the food, is the [food bank] system geared up to deal with the climate we live in?"

What both MTP and Food Aware have realised, in their different ways, is that through food they can begin to provide personalised support to clients by drawing together their diverse wider needs, whether debt, housing, health or joblessness, in a way that goes beyond the conventional "sticking plaster" food bank approach. Goodway has witnessed many clients' progress, their confidence slowly building. "Quite a few" get jobs, he says.

The number of people on MTP's books has remained stable, not because demand has slowed but because the charity has reached the limit – about 700 cases – of what it can feasibly do with available resources. It has had to close some cases and turn away referred clients on the basis that they are "too well off".

Goodway has calculated that the amount of food needed to address food poverty in Bristol runs into millions of kilos. MTP's annual food collection, donated by churches, schools, the public and surplus food distributors is a tiny fraction of that. To scale up, even to meet a tiny bit more of that need, would require proper investment and management.

Gibbons, too, is realistic about demand and supply: he says local authorities and NHS commissioners have to be involved in funding the "post-food bank" approach. At least £100,000 is needed for Food Aware to meet demand that he describes as "overwhelming". But even that figure – at least six times its current grant-funding level – may not be enough as welfare changes hit home.

MTP is a member of the Centre for Social Justice, the poverty thinktank set up by Conservative politician Iain Duncan Smith, now the work and pensions secretary. But Goodway is increasingly detached from what he sees as CSJ's refusal to acknowledge the malign and perverse effect of social security cuts.

He has been trying unsuccessfully to get a meeting with Duncan Smith to explain to him his belief, informed by his everyday experiences at MTP, that the sheer scale of welfare cuts "make it harder for people to get into work", while the "skiver-striver" rhetoric from politicians has had a toxic, demoralising effect on his clients.

Goodway, the former boss of a media company, is perhaps an unlikely activist, but his experiences have made him more determined to highlight the risks of austerity. "I'm not politically motivated, or a member of a political party. I'm political in so far as the system has to change."

• This article was amended on 19 June 2013. An earlier version referred to "a recent Greater London Authority investigation into food poverty". The investigation was by the London assembly.

• This article was amended on 25 June 2013 to correct the name of the Doncaster church to St James.